-
1
-
-
84889330859
-
-
Wiley, New York, 10.1002/9783527618958for a discussion of kinetic theory including the modern correlation function approach.
-
G. Mazenko, Nonequilibrium Statistical Mechanics (Wiley, New York, 2006) 10.1002/9783527618958
-
(2006)
Nonequilibrium Statistical Mechanics
-
-
Mazenko, G.1
-
2
-
-
77953448975
-
-
Brownian motion is interpreted here as a dynamics that is noise driven. For an individual Brownian particle located at position R acted upon by noise η, one has R =η. This is clearly the continuum version of the random-walk problem.Field theory methods enter via the techniques for organizing self-consistent perturbation theory in terms of the roles of external fields, cumulants, and irreducible vertex functions. The fundamental reference is
-
Brownian motion is interpreted here as a dynamics that is noise driven. For an individual Brownian particle located at position R acted upon by noise η, one has R = η. This is clearly the continuum version of the random-walk problem.
-
-
-
-
3
-
-
0003440273
-
-
4th ed. (Clarendon, Oxford, 10.1093/acprof:oso/9780198509233.001.0001We henceforth refer to this reference as JZJ.Kinetic theory's early history is briefly reviewed in
-
J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, 4th ed. (Clarendon, Oxford, 2002). 10.1093/acprof:oso/9780198509233.001.0001
-
(2002)
Quantum Field Theory and Critical Phenomena
-
-
Zinn-Justin, J.1
-
5
-
-
77953454761
-
-
By self-consistency we mean that the correlation and response functions of interest satisfy kinetic equations where the collision kernels, self-energies, or memory functions can be expressed in terms of the full physical correlation and response functions. In the areas of critical dynamics and the liquid-glass transition, this is essential in generating symmetry breaking solutions. This can generate a multiplicity of solutions not available in bare perturbation theory.
-
By self-consistency we mean that the correlation and response functions of interest satisfy kinetic equations where the collision kernels, self-energies, or memory functions can be expressed in terms of the full physical correlation and response functions. In the areas of critical dynamics and the liquid-glass transition, this is essential in generating symmetry breaking solutions. This can generate a multiplicity of solutions not available in bare perturbation theory.
-
-
-
-
6
-
-
77953453625
-
-
Boltzmann's Stosszahlansatz is a prime example of a decoupling approximation. In the collision integral of the Boltzmann equation he assumed that the two-particle distribution function factorizes into a product of two one-particle distributions, f2 → f1 f1.
-
Boltzmann's Stosszahlansatz is a prime example of a decoupling approximation. In the collision integral of the Boltzmann equation he assumed that the two-particle distribution function factorizes into a product of two one-particle distributions, f 2 → f 1 f 1.
-
-
-
-
8
-
-
0002931947
-
-
Pergamon, New York
-
S. G. Brush, Kinetic Theory (Pergamon, New York, 1966), Vol. 2.
-
(1966)
Kinetic Theory
, vol.2
-
-
Brush, S.G.1
-
10
-
-
0642359859
-
-
10.1016/0375-9601(70)90009-5;
-
K. Kawasaki, Phys. Lett. A 32, 379 (1970) 10.1016/0375-9601(70)90009-5
-
(1970)
Phys. Lett. A
, vol.32
, pp. 379
-
-
Kawasaki, K.1
-
11
-
-
33646377589
-
-
10.1143/PTP.45.1691;
-
K. Kawasaki, Prog. Theor. Phys. 45, 1691 (1971) 10.1143/PTP.45.1691
-
(1971)
Prog. Theor. Phys.
, vol.45
, pp. 1691
-
-
Kawasaki, K.1
-
12
-
-
0942290048
-
-
10.1016/0370-1573(75)90019-8; and dynamic critical phenomena:
-
Y. Pomeau and P. Resibois, Phys. Rep. 19, 63 (1975) 10.1016/0370-1573(75) 90019-8
-
(1975)
Phys. Rep.
, vol.19
, pp. 63
-
-
Pomeau, Y.1
Resibois, P.2
-
13
-
-
36849137696
-
-
10.1063/1.1732502;
-
M. Fixman, J. Chem. Phys. 36, 310 (1962) 10.1063/1.1732502
-
(1962)
J. Chem. Phys.
, vol.36
, pp. 310
-
-
Fixman, M.1
-
14
-
-
0002196716
-
-
10.1103/PhysRev.166.89;
-
L. P. Kadanoff and J. Swift, Phys. Rev. 166, 89 (1968) 10.1103/PhysRev.166.89
-
(1968)
Phys. Rev.
, vol.166
, pp. 89
-
-
Kadanoff, L.P.1
Swift, J.2
-
15
-
-
33646489567
-
-
10.1016/0003-4916(70)90375-1The role of mode-coupling theory has been characterized by
-
K. Kawasaki, Ann. Phys. (N.Y.) 61, 1 (1970). 10.1016/0003-4916(70)90375-1
-
(1970)
Ann. Phys. (N.Y.)
, vol.61
, pp. 1
-
-
Kawasaki, K.1
-
16
-
-
0035796146
-
-
10.1209/epl/i2001-00395-x
-
E. Zaccarelli, G. Foffi, F. Sciortino, P. Tartaglia, and K. A. Dawson, Europhys. Lett. 55, 157 (2001). 10.1209/epl/i2001-00395-x
-
(2001)
Europhys. Lett.
, vol.55
, pp. 157
-
-
Zaccarelli, E.1
Foffi, G.2
Sciortino, F.3
Tartaglia, P.4
Dawson, K.A.5
-
17
-
-
77953458551
-
-
In the weak supercooling regime, detailed predictions for the space and time dependence of the long-time decay of density correlations have been formulated using the ideal mode-coupling theory MCT, one of the first approaches to identify the existence of the crossover temperature. The agreement of MCT predictions with experimental findings and molecular dynamics simulations both for atomic and molecular models supports the view that MCT is indeed able to describe the slow dynamics in weak supercooled states.
-
In the weak supercooling regime, detailed predictions for the space and time dependence of the long-time decay of density correlations have been formulated using the ideal mode-coupling theory MCT, one of the first approaches to identify the existence of the crossover temperature. The agreement of MCT predictions with experimental findings and molecular dynamics simulations both for atomic and molecular models supports the view that MCT is indeed able to describe the slow dynamics in weak supercooled states.
-
-
-
-
18
-
-
0003625787
-
-
in edited by J. P. Hansen, D. Levesque, and J. Zinn-Justin (North-Holland, Amsterdam
-
W. Goetze, in Liquids, Freezing, and Glass Transition, edited by, J. P. Hansen,,, D. Levesque,, and, J. Zinn-Justin, (North-Holland, Amsterdam, 1991)
-
(1991)
Liquids, Freezing, and Glass Transition
-
-
Goetze, W.1
-
19
-
-
13144302948
-
Mode-coupling theory and the glass transition in supercooled liquids
-
DOI 10.1103/RevModPhys.76.785
-
and S. Das, Rev. Mod. Phys. 76, 785 (2004). 10.1103/RevModPhys.76.785 (Pubitemid 40180192)
-
(2004)
Reviews of Modern Physics
, vol.76
, Issue.3 I
, pp. 785-851
-
-
Das, S.P.1
-
20
-
-
77953405898
-
-
The important idea of an ergodic-nonergodic (ENE) transition exists independent of whether it conforms to the details of mode-coupling theory. The defining property of an ENE transition is that as a function of a control parameter there is crossover from an ergodic phase [ limt→ Gρρ (q,t ) =0 ] to a region with nonergodic kinetics [ limt→ Gρρ (q,t ) = A2 (q) >0 ]. Kinetic theory led to the early development of MCT. See for example
-
The important idea of an ergodic-nonergodic (ENE) transition exists independent of whether it conforms to the details of mode-coupling theory. The defining property of an ENE transition is that as a function of a control parameter there is crossover from an ergodic phase [lim t → G ρ ρ (q, t) = 0] to a region with nonergodic kinetics [lim t → G ρ ρ (q, t) = A 2 (q) > 0].
-
-
-
-
21
-
-
0000451148
-
-
10.1103/PhysRevA.7.209;
-
G. F. Mazenko, Phys. Rev. A 7, 209 (1973) 10.1103/PhysRevA.7.209
-
(1973)
Phys. Rev. A
, vol.7
, pp. 209
-
-
Mazenko, G.F.1
-
22
-
-
0042869520
-
-
10.1088/0022-3719/12/21/005;
-
L. Sjogren and A. Sjolander, J. Phys. C 12, 4369 (1979) 10.1088/0022-3719/12/21/005
-
(1979)
J. Phys. C
, vol.12
, pp. 4369
-
-
Sjogren, L.1
Sjolander, A.2
-
23
-
-
4243857431
-
-
10.1103/PhysRevA.29.2765;
-
E. Leutheusser, Phys. Rev. A 29, 2765 (1984) 10.1103/PhysRevA.29.2765
-
(1984)
Phys. Rev. A
, vol.29
, pp. 2765
-
-
Leutheusser, E.1
-
25
-
-
77953393365
-
-
More recent efforts to make microscopic contact (derive) mode-coupling theory are reflected in Ref. where they comment: "Despite its remarkable practical success, the presence of apparently uncontrolled approximations in the derivation of the MCT equations makes it difficult to gain insights into possible improvements of the theory. The aim of this paper is to present a new derivation of the ideal MCT equations, starting from the microscopic equations for the evolution of the density (Newtons equations) and writing them as a linear generalized Langevin equation. A formally exact expression for the memory kernel is derived and, on making the approximation that the noise in the Langevin equation is Gaussian, the standard MCT equations are obtained. Note that the proposition of Gaussian noise implies that the density fluctuations are also Gaussian."; Similarly in
-
More recent efforts to make microscopic contact (derive) mode-coupling theory are reflected in Ref. where they comment: "Despite its remarkable practical success, the presence of apparently uncontrolled approximations in the derivation of the MCT equations makes it difficult to gain insights into possible improvements of the theory. The aim of this paper is to present a new derivation of the ideal MCT equations, starting from the microscopic equations for the evolution of the density (Newtons equations) and writing them as a linear generalized Langevin equation. A formally exact expression for the memory kernel is derived and, on making the approximation that the noise in the Langevin equation is Gaussian, the standard MCT equations are obtained. Note that the proposition of Gaussian noise implies that the density fluctuations are also Gaussian."
-
-
-
-
26
-
-
42749101723
-
-
10.1103/PhysRevE.67.061116the authors make contact with MCT using a decoupling approximation which treats the density as a gaussian variable. Their characterization of the status of the theory is: "Although successful, the standard mode-coupling approximation has not been obtained in a systematic and straightforward fashion. A simple understanding of mode-coupling effects and their validity for describing low-temperature dynamics is still lacking. In this paper, we explore an alternative route to obtaining ideal mode-coupling equations via the direct Gaussian factorization of the multiple-point correlation function in the memory kernel."Field theory models for the glass transition are treated in
-
J. Wu and J. Cao, Phys. Rev. E 67, 061116 (2003) 10.1103/PhysRevE.67. 061116
-
(2003)
Phys. Rev. e
, vol.67
, pp. 061116
-
-
Wu, J.1
Cao, J.2
-
27
-
-
33746725529
-
-
10.1103/PhysRevA.34.2265;
-
S. P. Das and G. F. Mazenko, Phys. Rev. A 34, 2265 (1986) 10.1103/PhysRevA.34.2265
-
(1986)
Phys. Rev. A
, vol.34
, pp. 2265
-
-
Das, S.P.1
Mazenko, G.F.2
-
28
-
-
61949358366
-
-
10.1103/PhysRevE.79.021504;
-
S. P. Das and G. F. Mazenko, Phys. Rev. E 79, 021504 (2009) 10.1103/PhysRevE.79.021504
-
(2009)
Phys. Rev. e
, vol.79
, pp. 021504
-
-
Das, S.P.1
Mazenko, G.F.2
-
29
-
-
52649130798
-
-
10.1103/PhysRevE.78.031123;
-
G. F. Mazenko, Phys. Rev. E 78, 031123 (2008) 10.1103/PhysRevE.78.031123
-
(2008)
Phys. Rev. e
, vol.78
, pp. 031123
-
-
Mazenko, G.F.1
-
30
-
-
0030597463
-
-
10.1088/0305-4470/29/24/001;
-
D. S. Dean, J. Phys. A 29, L613 (1996) 10.1088/0305-4470/29/24/001
-
(1996)
J. Phys. A
, vol.29
, pp. 613
-
-
Dean, D.S.1
-
32
-
-
18744415431
-
-
10.1088/0305-4470/38/20/L03;
-
K. Miyazaki and D. R. Reichman, J. Phys. A 38, L343 (2005) 10.1088/0305-4470/38/20/L03
-
(2005)
J. Phys. A
, vol.38
, pp. 343
-
-
Miyazaki, K.1
Reichman, D.R.2
-
36
-
-
40849126943
-
-
10.1016/j.nuclphysb.2007.11.039In the case of Smoluchowski dynamics it has been shown by
-
A. Crisanti, Nucl. Phys. B 796, 425 (2008). 10.1016/j.nuclphysb.2007.11. 039
-
(2008)
Nucl. Phys. B
, vol.796
, pp. 425
-
-
Crisanti, A.1
-
37
-
-
0000190673
-
-
10.1016/0378-4371(87)90176-2that the memory function can be rewritten in terms of an irreducible memory function. This argument was generalized by
-
B. Cichocki and W. Hess, Physica A 141, 475 (1987) 10.1016/0378-4371(87) 90176-2
-
(1987)
Physica A
, vol.141
, pp. 475
-
-
Cichocki, B.1
Hess, W.2
-
38
-
-
0000858933
-
-
10.1016/0378-4371(95)00012-VFor our purposes here the point is that there is not a unique memory function form at lowest nontrivial order in perturbation theory. Thus for a variety of models one can rearrange perturbation theory to have model which supports an ENE transition. One must go to higher order to check the self-consistency of an ENE transition.The theory we develop here has great potential for treating higher-order correlation functions. The reason why this is interesting is because there has been a significant amount of work associated with the concept of dynamic heterogeneity, e.g.
-
K. Kawasaki, Physica A 215, 61 (1995). 10.1016/0378-4371(95)00012-V
-
(1995)
Physica A
, vol.215
, pp. 61
-
-
Kawasaki, K.1
-
39
-
-
0034273127
-
-
-ν, and =T- Tc There is both experimental and numerical support for this hypothesis [See
-
S. C. Glotzer, J. Non-Cryst. Solids 274, 342 (2000). 10.1016/S0022- 3093(00)00225-8
-
(2000)
J. Non-Cryst. Solids
, vol.274
, pp. 342
-
-
Glotzer, S.C.1
-
40
-
-
42749105310
-
-
10.1103/PhysRevE.69.020201;
-
L. Berthier, Phys. Rev. E 69, 020201 (2004).] 10.1103/PhysRevE.69.020201
-
(2004)
Phys. Rev. e
, vol.69
, pp. 020201
-
-
Berthier, L.1
-
41
-
-
77953467890
-
-
note
-
Our interest here is the theoretical work of Biroli and Bouchaud in Ref.. Biroli and Bouchaud sketched a field theoretical calculation of C 4 (r, t) compatible with mode-coupling theory. This calculation leads to a diverging length scale at the ideal glass transition. They find an upper critical dimension of six in their calculation. This work is very provocative since at the level of the dynamic structure factor it is well known that MCT does not contain a large length as one approaches the ergodic-nonergodic transition. Biroli and Bouchaud suggest that one must dig deeper into the theory, look at the four-point quantity C 4, to find this diverging length. However this calculation paints a picture rather than gives the results of a rigorous calculation. One of the goals for the theory developed here is to connect the collective behaviors found at the two-point level to the behavior of the three- and four-point density correlation functions.
-
-
-
-
42
-
-
4243948548
-
-
10.1103/PhysRevB.11.4077
-
S. Ma and G. Mazenko, Phys. Rev. B 11, 4077 (1975). 10.1103/PhysRevB.11. 4077
-
(1975)
Phys. Rev. B
, vol.11
, pp. 4077
-
-
Ma, S.1
Mazenko, G.2
-
43
-
-
77953406977
-
-
Purely dissipative Langevin systems have long been used to describe the order parameter dynamics in magnetic and superfluid systems. There is a balance between the force due to the gradient of the effective free energy and the persistent noise in a thermalized system.The path-integral description of classical Newtonian dynamics is developed in a series of sophisticated papers by Gozzi and collaborators.
-
Purely dissipative Langevin systems have long been used to describe the order parameter dynamics in magnetic and superfluid systems. There is a balance between the force due to the gradient of the effective free energy and the persistent noise in a thermalized system.
-
-
-
-
44
-
-
0000223868
-
-
10.1016/0370-2693(88)90611-9;
-
E. Gozzi, Phys. Lett. B 201, 525 (1988) 10.1016/0370-2693(88)90611-9
-
(1988)
Phys. Lett. B
, vol.201
, pp. 525
-
-
Gozzi, E.1
-
48
-
-
77953410818
-
-
Their interest is in the symmetry structure of the coupled spaces when one allows the classical system to be coupled to a system of ghost fermions. See below.
-
Their interest is in the symmetry structure of the coupled spaces when one allows the classical system to be coupled to a system of ghost fermions. See below.
-
-
-
-
49
-
-
77953370677
-
-
Fokker-Planck dynamics in modern terms has two complementary meanings. For systems of particles governed by phase-space coordinates and with a damping component in their dynamics the equation satisfied by the associated probability distribution is called a Fokker-Planck equation. As pointed out by Lax and Zwanzig, the associated equation of motion for the phase-space variables is referred to as a Langevin equation. Subsequently the transition generally from a stochastic equation of motion to the time evolution of the associated probability description is called going from a generalized Langevin equation description to generalized Fokker-Planck description. See Ref.. Smoluchowski dynamics has become identified with the over damped kinetics in colloidal systems where the momenta become equilibrated much faster than the positions and one has a dynamics which is subsequently organized in terms of the positions. This is the point of view taken by
-
Fokker-Planck dynamics in modern terms has two complementary meanings. For systems of particles governed by phase-space coordinates and with a damping component in their dynamics the equation satisfied by the associated probability distribution is called a Fokker-Planck equation. As pointed out by Lax and Zwanzig, the associated equation of motion for the phase-space variables is referred to as a Langevin equation. Subsequently the transition generally from a stochastic equation of motion to the time evolution of the associated probability description is called going from a generalized Langevin equation description to generalized Fokker-Planck description. See Ref..
-
-
-
-
50
-
-
84977586068
-
-
10.1002/andp.19053220806in his seminal treatment of Brownian motion;
-
A. Einstein, Ann. Phys. 322, 549 (1905) 10.1002/andp.19053220806
-
(1905)
Ann. Phys.
, vol.322
, pp. 549
-
-
Einstein, A.1
-
56
-
-
34547275473
-
-
10.1016/S0031-8914(40)90098-2; This field has been famously reviewed by
-
H. Kramers, Physica 7, 284 (1940) 10.1016/S0031-8914(40)90098-2
-
(1940)
Physica
, vol.7
, pp. 284
-
-
Kramers, H.1
-
58
-
-
0002121327
-
-
10.1103/RevModPhys.15.1;
-
S. Chandrasekhar, Rev. Mod. Phys. 15, 1 (1943) 10.1103/RevModPhys.15.1
-
(1943)
Rev. Mod. Phys.
, vol.15
, pp. 1
-
-
Chandrasekhar, S.1
-
59
-
-
0002783430
-
-
10.1103/RevModPhys.32.25;
-
M. Lax, Rev. Mod. Phys. 32, 25 (1960) 10.1103/RevModPhys.32.25
-
(1960)
Rev. Mod. Phys.
, vol.32
, pp. 25
-
-
Lax, M.1
-
60
-
-
0000514214
-
-
10.1016/0022-3697(60)90237-7;
-
M. Lax, J. Phys. Chem. Solids 14, 248 (1960) 10.1016/0022-3697(60)90237-7
-
(1960)
J. Phys. Chem. Solids
, vol.14
, pp. 248
-
-
Lax, M.1
-
61
-
-
33847758506
-
-
10.1103/RevModPhys.38.541; The modern, linear response, correlation function treatment of Brownian motion is due to
-
M. Lax, Rev. Mod. Phys. 38, 541 (1966) 10.1103/RevModPhys.38.541
-
(1966)
Rev. Mod. Phys.
, vol.38
, pp. 541
-
-
Lax, M.1
-
62
-
-
36149022800
-
-
10.1103/PhysRev.131.2381; The associated Fokker-Planck description for more than one Brownian particle was developed by
-
J. L. Lebowitz and E. Rubin, Phys. Rev. 131, 2381 (1963) 10.1103/PhysRev.131.2381
-
(1963)
Phys. Rev.
, vol.131
, pp. 2381
-
-
Lebowitz, J.L.1
Rubin, E.2
-
63
-
-
0000274247
-
-
10.1063/1.1675379; The extension of the multiple particle Fokker-Planck description into the over damped Smoluchowski regime was carried out by
-
J. Deutch and I. Oppenheim, J. Chem. Phys. 54, 3547 (1971) 10.1063/1.1675379
-
(1971)
J. Chem. Phys.
, vol.54
, pp. 3547
-
-
Deutch, J.1
Oppenheim, I.2
-
65
-
-
84949208823
-
-
10.1002/9780470143605.ch17using results reviewed by M. Lax, showed the equivalence of the Fokker-Planck and Langevin descriptions; Important papers treating the many-particle Smoluchowski dynamics system include
-
R. Zwanzig, Adv. Chem. Phys. 15, 325 (1969) 10.1002/9780470143605.ch17
-
(1969)
Adv. Chem. Phys.
, vol.15
, pp. 325
-
-
Zwanzig, R.1
-
66
-
-
0000672076
-
-
10.1063/1.431957;
-
B. J. Ackerson, J. Chem. Phys. 64, 242 (1976) 10.1063/1.431957
-
(1976)
J. Chem. Phys.
, vol.64
, pp. 242
-
-
Ackerson, B.J.1
-
67
-
-
42849083834
-
-
10.1016/0378-4371(79)90052-9A modern introduction to the equilibrium theory of fluids is given by
-
W. Dieterich and I. Peschel, Physica A 95, 208 (1979). 10.1016/0378-4371(79)90052-9
-
(1979)
Physica A
, vol.95
, pp. 208
-
-
Dieterich, W.1
Peschel, I.2
-
69
-
-
0002457898
-
-
10.1063/1.1749933;
-
J. E. Mayer, J. Chem. Phys. 5, 67 (1937) 10.1063/1.1749933
-
(1937)
J. Chem. Phys.
, vol.5
, pp. 67
-
-
Mayer, J.E.1
-
73
-
-
18344374952
-
-
and, 10.1063/1.1750964. This work showed how a primitive density expansion could be used in dense systems.
-
and E. Montroll and J. E. Mayer, J. Chem. Phys. 9, 626 (1941) 10.1063/1.1750964
-
(1941)
J. Chem. Phys.
, vol.9
, pp. 626
-
-
Montroll, E.1
Mayer, J.E.2
-
74
-
-
33646471468
-
-
10.1063/1.1749657;
-
J. G. Kirkwood, J. Chem. Phys. 3, 300 (1935) 10.1063/1.1749657
-
(1935)
J. Chem. Phys.
, vol.3
, pp. 300
-
-
Kirkwood, J.G.1
-
75
-
-
77953428665
-
-
Actualités Sci. Ind.
-
J. Yvon, Actualités Sci. Ind. 203 (1935)
-
(1935)
, vol.203
-
-
Yvon, J.1
-
79
-
-
0039964356
-
-
10.1073/pnas.42.3.122;
-
G. W. Ford and G. E. Uhlenbeck, PNAS 42, 122 (1956) 10.1073/pnas.42.3.122
-
(1956)
PNAS
, vol.42
, pp. 122
-
-
Ford, G.W.1
Uhlenbeck, G.E.2
-
80
-
-
0008925981
-
-
10.1073/pnas.42.8.529;
-
G. W. Ford and G. E. Uhlenbeck, PNAS 42, 529 (1956) 10.1073/pnas.42.8.529
-
(1956)
PNAS
, vol.42
, pp. 529
-
-
Ford, G.W.1
Uhlenbeck, G.E.2
-
81
-
-
0013333654
-
-
10.1073/pnas.43.1.163;
-
G. W. Ford and G. E. Uhlenbeck, PNAS 43, 163 (1957) 10.1073/pnas.43.1.163
-
(1957)
PNAS
, vol.43
, pp. 163
-
-
Ford, G.W.1
Uhlenbeck, G.E.2
-
83
-
-
77952457788
-
-
10.1143/JPSJ.12.864;
-
K. Hiroike, J. Phys. Soc. Jpn. 12, 864 (1957) 10.1143/JPSJ.12.864
-
(1957)
J. Phys. Soc. Jpn.
, vol.12
, pp. 864
-
-
Hiroike, K.1
-
88
-
-
0001270570
-
-
10.1063/1.1743985;
-
E. Meeron, J. Chem. Phys. 27, 1238 (1957) 10.1063/1.1743985
-
(1957)
J. Chem. Phys.
, vol.27
, pp. 1238
-
-
Meeron, E.1
-
89
-
-
51249193561
-
-
10.1007/BF02824240;
-
J. Yvon, Nuovo Cimento 9, 144 (1958) 10.1007/BF02824240
-
(1958)
Nuovo Cimento
, vol.9
, pp. 144
-
-
Yvon, J.1
-
91
-
-
36849132997
-
-
10.1063/1.1703949Functional methods were introduced by
-
C. De Dominicis, J. Math. Phys. 4, 255 (1963). 10.1063/1.1703949
-
(1963)
J. Math. Phys.
, vol.4
, pp. 255
-
-
De Dominicis, C.1
-
94
-
-
0003440479
-
-
in edited by H. L. Frisch and J. L. Lebowitz (Benjamin, New York
-
J. K. Percus, in The Equilibrium Theory of Classical Fluids, edited by, H. L. Frisch, and, J. L. Lebowitz, (Benjamin, New York, 1964).
-
(1964)
The Equilibrium Theory of Classical Fluids
-
-
Percus, J.K.1
-
95
-
-
7544220942
-
-
10.1103/PhysRev.110.1See the discussion of
-
J. K. Percus and G. S. Yevick, Phys. Rev. 110, 1 (1958). 10.1103/PhysRev.110.1
-
(1958)
Phys. Rev.
, vol.110
, pp. 1
-
-
Percus, J.K.1
Yevick, G.S.2
-
96
-
-
0345045853
-
-
10.1088/0034-4885/28/1/306
-
J. S. Rowlinson, Rep. Prog. Phys. 28, 169 (1965). 10.1088/0034-4885/28/1/ 306
-
(1965)
Rep. Prog. Phys.
, vol.28
, pp. 169
-
-
Rowlinson, J.S.1
-
97
-
-
77953383548
-
-
The Ornstein-Zernike relation connects the radial distribution function to the direct correlation function. See L. S. Ornstein and F. Zernike in Ref.. Early treatments of classical field dynamics include
-
The Ornstein-Zernike relation connects the radial distribution function to the direct correlation function. See L. S. Ornstein and F. Zernike in Ref..
-
-
-
-
98
-
-
36149021145
-
-
10.1103/PhysRev.91.1505;
-
L. Onsager and S. Machlup, Phys. Rev. 91, 1505 (1953) 10.1103/PhysRev.91.1505
-
(1953)
Phys. Rev.
, vol.91
, pp. 1505
-
-
Onsager, L.1
MacHlup, S.2
-
99
-
-
84891211683
-
-
10.1017/S0022112059000362;
-
R. H. Kraichnan, J. Fluid Mech. 5, 497 (1959) 10.1017/S0022112059000362
-
(1959)
J. Fluid Mech.
, vol.5
, pp. 497
-
-
Kraichnan, R.H.1
-
100
-
-
36849142102
-
-
10.1063/1.1724206;
-
R. H. Kraichnan, J. Math. Phys. 2, 124 (1961) 10.1063/1.1724206
-
(1961)
J. Math. Phys.
, vol.2
, pp. 124
-
-
Kraichnan, R.H.1
-
101
-
-
0004058443
-
-
10.1063/1.1703782;
-
R. H. Kraichnan, J. Math. Phys. 3, 205 (1962) 10.1063/1.1703782
-
(1962)
J. Math. Phys.
, vol.3
, pp. 205
-
-
Kraichnan, R.H.1
-
102
-
-
0001330827
-
-
10.1063/1.2746572;
-
R. H. Kraichnan, Phys. Fluids 7, 1723 (1964) 10.1063/1.2746572
-
(1964)
Phys. Fluids
, vol.7
, pp. 1723
-
-
Kraichnan, R.H.1
-
103
-
-
0012749240
-
-
10.1016/0003-4916(61)90056-2
-
H. W. Wyld, Jr., Ann. Phys. (N.Y.) 14, 143 (1961) 10.1016/0003-4916(61) 90056-2
-
(1961)
Ann. Phys. (N.Y.)
, vol.14
, pp. 143
-
-
Wyld, Jr.H.W.1
-
111
-
-
0001702607
-
-
10.1103/PhysRevA.11.2043;
-
U. Deker and F. Haake, Phys. Rev. A 11, 2043 (1975) 10.1103/PhysRevA.11. 2043
-
(1975)
Phys. Rev. A
, vol.11
, pp. 2043
-
-
Deker, U.1
Haake, F.2
-
112
-
-
0001380349
-
-
10.1088/0305-4470/8/9/011;
-
R. Phythian, J. Phys. A 8, 1423 (1975) 10.1088/0305-4470/8/9/011
-
(1975)
J. Phys. A
, vol.8
, pp. 1423
-
-
Phythian, R.1
-
113
-
-
0000336768
-
-
10.1088/0305-4470/9/2/012;
-
R. Phythian, J. Phys. A 9, 269 (1976) 10.1088/0305-4470/9/2/012
-
(1976)
J. Phys. A
, vol.9
, pp. 269
-
-
Phythian, R.1
-
114
-
-
4243311531
-
-
10.1103/PhysRevA.19.846;
-
U. Deker, Phys. Rev. A 19, 846 (1979) 10.1103/PhysRevA.19.846
-
(1979)
Phys. Rev. A
, vol.19
, pp. 846
-
-
Deker, U.1
-
116
-
-
5844235899
-
-
10.1007/BF01022182TDGL models were first studied using these new techniques by
-
R. V. Jensen, J. Stat. Phys. 25, 183 (1981). 10.1007/BF01022182
-
(1981)
J. Stat. Phys.
, vol.25
, pp. 183
-
-
Jensen, R.V.1
-
119
-
-
77953409731
-
-
- ψ̄ Mψ. See page 14 in JZJ. These are the ghost fermions and eliminating the determinant in favor of the fermions gives a coupled fermion-nonfermion system
-
- ψ̄ Mψ. See page 14 in JZJ. These are the ghost fermions and eliminating the determinant in favor of the fermions gives a coupled fermion-nonfermion system.
-
-
-
-
121
-
-
33646670440
-
-
10.1016/0550-3213(82)90538-7;
-
G. Parisi and N. Sourlas, Nucl. Phys. B 206, 321 (1982) 10.1016/0550-3213(82)90538-7
-
(1982)
Nucl. Phys. B
, vol.206
, pp. 321
-
-
Parisi, G.1
Sourlas, N.2
-
125
-
-
0001945666
-
-
10.1103/PhysRevD.28.1922;
-
E. Gozzi, Phys. Rev. D 28, 1922 (1983) 10.1103/PhysRevD.28.1922
-
(1983)
Phys. Rev. D
, vol.28
, pp. 1922
-
-
Gozzi, E.1
-
126
-
-
0007034258
-
-
10.1016/0550-3213(86)90592-4;
-
J. Zinn-Justin, Nucl. Phys. B 275, 135 (1986) 10.1016/0550-3213(86)90592- 4
-
(1986)
Nucl. Phys. B
, vol.275
, pp. 135
-
-
Zinn-Justin, J.1
-
127
-
-
34249972705
-
-
10.1007/BF01044231Once one has the ghost fields as described in Ref. in the generating functional for the field, one can search for symmetries of the action reflected in the whole space of the enlarged system. For a large class of stochastic systems one finds a supersymmetry which mixes fermionic and conventional field degrees of freedom. The key references are given in Ref.. One can go further in the development as indicated by the title "Ward Takahashi Identities and Fluctuation-Dissipation Theorem in a superspace formulation of the Langevin Equation" by
-
G. Munoz and W. S. Burgett, J. Stat. Phys. 56, 59 (1989). 10.1007/BF01044231
-
(1989)
J. Stat. Phys.
, vol.56
, pp. 59
-
-
Munoz, G.1
Burgett, W.S.2
-
129
-
-
0040640096
-
-
10.1088/0305-4470/17/10/016. By rewriting the total action in terms of a superfield one obtains a highly compact and symmetric action. This is discussed in some detail in JZJ. A strong cautionary note: Using supersymmetry arguments, it is possible demonstrate that one has dimensional reduction in the random field Ising model (RFIM). Thus the RFIM in four dimensions maps onto the pure Ising model in two dimensions. Unfortunately the four dimensional RFIM does not behave like the two-dimensional pure Ising model under conditions of equilibrium as measured by numerical studies. The explanation put forth by JZJ is that the physical solution of the field equations is a supersymmetry broken solution. For a discussion see
-
S. Chaturvedi, A. K. Kapoor and V. Srinivason, J. Phys. A 17, 2037 (1984) 10.1088/0305-4470/17/10/016
-
(1984)
J. Phys. A
, vol.17
, pp. 2037
-
-
Chaturvedi, S.1
Kapoor, A.K.2
Srinivason, V.3
-
130
-
-
0020722697
-
-
10.1063/1.333669See Onsager's Principle of Microscopic Reversibility and Supersymmetry
-
G. Grinstein, J. Appl. Phys. 55, 2371 (1984). 10.1063/1.333669
-
(1984)
J. Appl. Phys.
, vol.55
, pp. 2371
-
-
Grinstein, G.1
-
131
-
-
3543040920
-
-
10.1103/PhysRevD.30.1218;
-
E. Gozzi, Phys. Rev. D 30, 1218 (1984) 10.1103/PhysRevD.30.1218
-
(1984)
Phys. Rev. D
, vol.30
, pp. 1218
-
-
Gozzi, E.1
-
132
-
-
36149006492
-
-
10.1103/PhysRev.37.405;
-
L. Onsager, Phys. Rev. 37, 405 (1931) 10.1103/PhysRev.37.405
-
(1931)
Phys. Rev.
, vol.37
, pp. 405
-
-
Onsager, L.1
-
133
-
-
36149006040
-
-
10.1103/PhysRev.38.2265;
-
L. Onsager, Phys. Rev. 38, 2265 (1931) 10.1103/PhysRev.38.2265
-
(1931)
Phys. Rev.
, vol.38
, pp. 2265
-
-
Onsager, L.1
-
134
-
-
36149003004
-
-
10.1103/RevModPhys.17.343There are a collection of fluctuation theorems which appear to connect strongly nonequilibrium systems and which should have a counterpart in the particle systems discussed here. Outing
-
N. B. Casimir, Rev. Mod. Phys. 17, 343 (1945). 10.1103/RevModPhys.17.343
-
(1945)
Rev. Mod. Phys.
, vol.17
, pp. 343
-
-
Casimir, N.B.1
-
135
-
-
42749108175
-
-
10.1088/1742-5468/2006/08/P08001; The fluctuation theorem refers to a set of exact relations describing the statistical mechanics of systems away from equilibrium, generically expressed by the formula, P (+Σ ) /P (-Σ ) = eΣ where P (Σ) is the distribution of observed values of a quantity representing dissipation or entropy production. Such theorems have been worked out for a variety of nonequilibrium systems:
-
V. Y. Chernyak, M. Chertkov, and C. Jarzynski, J. Stat. Mech.: Theory Exp. (2006), P08001 10.1088/1742-5468/2006/08/P08001
-
J. Stat. Mech.: Theory Exp.
, vol.2006
, pp. 08001
-
-
Chernyak, V.Y.1
Chertkov, M.2
Jarzynski, C.3
-
139
-
-
0040120726
-
-
10.1088/0305-4470/31/16/003;
-
J. Kurchan, J. Phys. A 31, 3719 (1998) 10.1088/0305-4470/31/16/003
-
(1998)
J. Phys. A
, vol.31
, pp. 3719
-
-
Kurchan, J.1
-
143
-
-
4243754128
-
-
10.1103/PhysRevLett.78.2690;
-
C. Jarzynski, Phys. Rev. Lett. 78, 2690 (1997) 10.1103/PhysRevLett.78. 2690
-
(1997)
Phys. Rev. Lett.
, vol.78
, pp. 2690
-
-
Jarzynski, C.1
-
144
-
-
4244116139
-
-
10.1103/PhysRevE.56.5018;
-
C. Jarzynski, Phys. Rev. E 56, 5018 (1997) 10.1103/PhysRevE.56.5018
-
(1997)
Phys. Rev. e
, vol.56
, pp. 5018
-
-
Jarzynski, C.1
-
145
-
-
0032023968
-
-
10.1023/A:1023208217925;
-
G. E. Crooks, J. Stat. Phys. 90, 1481 (1998) 10.1023/A:1023208217925
-
(1998)
J. Stat. Phys.
, vol.90
, pp. 1481
-
-
Crooks, G.E.1
-
146
-
-
0000765761
-
-
10.1103/PhysRevE.60.2721. These theorems should have counterparts in the theories of particles developed here. There is the rather amazing claim by
-
G. E. Crooks, Phys. Rev. E 60, 2721 (1999) 10.1103/PhysRevE.60.2721
-
(1999)
Phys. Rev. e
, vol.60
, pp. 2721
-
-
Crooks, G.E.1
-
147
-
-
77953439846
-
-
e-print arXiv:0711.2059, that these theorems are a consequence of an underlying supersymmetry. "This supersymmetry in turn allows one to generate the fluctuation-dissipation theorem to far from equilibrium situations.
-
K. Mallick, M. Moshe, and H. Orland, e-print arXiv:0711.2059
-
-
-
Mallick, K.1
Moshe, M.2
Orland, H.3
-
148
-
-
0001052368
-
-
10.1007/BF01011780
-
H. Rose, J. Stat. Phys. 20, 415 (1979). 10.1007/BF01011780
-
(1979)
J. Stat. Phys.
, vol.20
, pp. 415
-
-
Rose, H.1
-
149
-
-
77953381054
-
-
). If the density is a Gaussian variable, then W [u] = dd x1 dd x2 u ( x1 ) G ( x1 - x2 ) u ( x2 ).
-
). If the density is a Gaussian variable, then W [u] = d d x 1 d d x 2 u (x 1) G (x 1 - x 2) u (x 2).
-
-
-
-
150
-
-
77953452108
-
-
JZJ gives an argument on page 63 for choosing θ (0) =1/2.
-
JZJ gives an argument on page 63 for choosing θ (0) = 1 / 2.
-
-
-
-
151
-
-
77953451518
-
-
This is just the functional generalization of the standard matrix identity exp [ 1 2 ij Kij δ2 δ hi δ hj ] e i hi i = e i hi i exp [ 1 2 ij Kij i j ].
-
This is just the functional generalization of the standard matrix identity exp [1 2 i j K i j δ 2 δ h i δ h j] e i h i i = e i h i i exp [1 2 i j K i j i j].
-
-
-
-
153
-
-
33646515721
-
-
10.1103/PhysRev.124.287;
-
G. Baym and L. P. Kadanoff, Phys. Rev. 124, 287 (1961) 10.1103/PhysRev.124.287
-
(1961)
Phys. Rev.
, vol.124
, pp. 287
-
-
Baym, G.1
Kadanoff, L.P.2
-
154
-
-
33748957360
-
-
10.1103/PhysRev.127.1391
-
G. Baym, Phys. Rev. 127, 1391 (1962). 10.1103/PhysRev.127.1391
-
(1962)
Phys. Rev.
, vol.127
, pp. 1391
-
-
Baym, G.1
-
155
-
-
77953379909
-
-
In different contexts the kinetic kernels have different names. In the field theory protocol the kernels are typically called self-energies, in the kinetic theory protocol, where the analysis is in terms of retarded quantities, the kernel is called a memory function, and in the general case it can be called a dynamic direct correlation function.
-
In different contexts the kinetic kernels have different names. In the field theory protocol the kernels are typically called self-energies, in the kinetic theory protocol, where the analysis is in terms of retarded quantities, the kernel is called a memory function, and in the general case it can be called a dynamic direct correlation function.
-
-
-
-
156
-
-
44449157712
-
-
10.1088/1751-8113/41/23/235002
-
R. Velenich, C. Chamon, L. Cugliandolo, and D. Kreimer, J. Phys. A 41, 235002 (2008). 10.1088/1751-8113/41/23/235002
-
(2008)
J. Phys. A
, vol.41
, pp. 235002
-
-
Velenich, R.1
Chamon, C.2
Cugliandolo, L.3
Kreimer, D.4
-
157
-
-
0000137755
-
-
10.1103/PhysRev.108.590see the introduction and Appendix D.The classical operator formalism for dynamics was introduced by
-
W. Kohn and J. Luttinger, Phys. Rev. 108, 590 (1957) 10.1103/PhysRev.108. 590
-
(1957)
Phys. Rev.
, vol.108
, pp. 590
-
-
Kohn, W.1
Luttinger, J.2
-
159
-
-
0009183042
-
-
10.1073/pnas.18.3.255Quantum many-body theory is close to the development here. In particular the functional formulation of
-
H. Koopman and J. von Neumann, Proc. Natl. Acad. Sci. U.S.A. 18, 255 (1932). 10.1073/pnas.18.3.255
-
(1932)
Proc. Natl. Acad. Sci. U.S.A.
, vol.18
, pp. 255
-
-
Koopman, H.1
Von Neumann, J.2
-
160
-
-
0043141866
-
-
10.1103/PhysRev.115.1342; as popularized by
-
P. C. Martin and J. Schwinger, Phys. Rev. 115, 1342 (1959) 10.1103/PhysRev.115.1342
-
(1959)
Phys. Rev.
, vol.115
, pp. 1342
-
-
Martin, P.C.1
Schwinger, J.2
-
162
-
-
77953453291
-
-
Quantum Thermal Green's functions must satisfy Kubo-Martin-Schwinger Boundary Conditions, see references in Ref.. See the discussion of this result for the statics in
-
Quantum Thermal Green's functions must satisfy Kubo-Martin-Schwinger Boundary Conditions, see references in Ref..
-
-
-
-
163
-
-
77953458550
-
-
in edited by H. L. Frisch and J. L. Lebowitz (Benjamin, New York
-
J. K. Percus, in The Equilibrium Theory of Classical Fluids, edited by, H. L. Frisch, and, J. L. Lebowitz, (Benjamin, New York, 1964), p. II-72.
-
(1964)
The Equilibrium Theory of Classical Fluids
, pp. 72
-
-
Percus, J.K.1
-
164
-
-
77953443965
-
-
The decay time in the problem is inversely proportional to the static structure factor. This clearly, away from q=0, leads to a slowing down near the first structure factor peak.There is an exact solution to the Percus-Yevick approximation for hard spheres. See the discussion in
-
The decay time in the problem is inversely proportional to the static structure factor. This clearly, away from q = 0, leads to a slowing down near the first structure factor peak.
-
-
-
-
165
-
-
85014175361
-
-
3rd ed., (Academic Press, New York
-
J.-P. Hansen and I. R. McDonald, Theory of Simple Liquids, 3rd ed., (Academic Press, New York, 2006), Chap..
-
(2006)
Theory of Simple Liquids
-
-
Hansen, J.-P.1
McDonald, I.R.2
-
166
-
-
0003625787
-
-
in edited by J. P. Hansen, D. Levesque, and J. Zinn-Justin (North-Holland, Amsterdam
-
W. Goetze in Liquids, Freezing, and Glass Transition, edited by, J. P. Hansen,,, D. Levesque,, and, J. Zinn-Justin, (North-Holland, Amsterdam, 1991).
-
(1991)
Liquids, Freezing, and Glass Transition
-
-
Goetze, W.1
|